NBER WORKING PAPER SERIES CAN IT BE JAPAN S SAVIOR. Fumio Hayashi Koji Nomura. Working Paper

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1 NBER WORKING PAPER SERIES CAN IT BE JAPAN S SAVIOR Fumio Hayashi Koji Nomura Working Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA November 2005 The research by the first author was supported by Grants-in-Aid for Scientific Research No administered by the Ministry of Education, Culture, Sports, Science, and Technology of the Japanese government. The views expressed herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of Economic Research by Fumio Hayashi and Koji Nomura. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including notice, is given to the source.

2 Can IT be Japan s Savior? Fumio Hayashi and Koji Nomura NBER Working Paper No November 2005 JEL No. E2, O4, O5 ABSTRACT This paper constructs a multi-sector model to take explicit account of the very sharp change in the relative price between non-it and IT goods. The model is calibrated to the Japanese economy, and its solution path from 1990 on is compared to Japan's macroeconomic performance in the 1990s. Compared to the one-sector analysis of Japan in the 1990s in Hayashi and Prescott (2002), our model does slightly better or just as well in accounting for Japan's output slump and does worse in accounting for the capital-output ratio. We also show that, to revive a 2% long-term growth in percapita GDP, Japan needs to direct 10% of private total hours to the IT sector. Fumio Hayashi Department of Economics University of Tokyo Tokyo , Japan and NBER Koji Nomura Keio Economic Observatory Keio University Mita, Tokyo , Japan

3 1. Introduction We live in a world where IT goods, such as computers and communications equipments, are continuously getting cheaper than the rest of the goods, at a relentless rate. As shown in Figure 1 for Japan, the relative price of IT goods to non-it goods in 2000 is one-eightieth of what it was in Figure 1: Relative Price of IT Goods in Terms of non-it Goods (year 2000 = 1) The rapid change in the relative price creates potentially serious problems for the one-good world of macroeconomics. The Hicks aggregation theorem, which allows a bundle of many goods to be treated as if it is a single good, is no longer valid. If the relative price of two goods is not constant, the utility function and the production function must have those two goods as separate arguments. One could invoke the theory of aggregation of heterogenous capital goods to have a single variable representing capital in the production function, but that single index, called the capital services index, is different from the simple sum of the capital stocks, as Jorgenson and Griliches (1967) first pointed out. For both the (aggregated) capital stock and the capital services index, the growth rate is a 1 See the text below for the definition of non-it and IT goods and how the relative price is calculated. 2

4 weighted averages of the growth rates of individual capital stocks. The weights are the value shares for the capital stock and the user-cost shares for capital services. The difference can be substantial when non-it capital and IT capital are to be aggregated. IT capital s user cost is much higher than non-it capital s because IT capital depreciates fast physically and in value. If the stock of IT capital is growing fast, the growth rate of capital services is higher than that of the capital stock by a substantial margin, even if IT capital s value share is small. In the growth accounting typically practiced in macroeconomics, the contribution of capital is measured by the growth rate of the aggregated capital stock. The TFP (total factor productivity) growth calculated by the macro growth accounting, therefore, confuses genuine TFP growth with the contribution of IT capital to the growth of capital services. Hayashi and Prescott (2002) showed that Japan s great stagnation in the 1990s is wellaccounted for by the standard neoclassical growth model with a TFP slowdown in the 1990s. Their one-good model, however, does not explicitly take into account the relentless decline in the relative price of IT goods. It is possible that the decline in TFP growth, which is the cornerstone of their analysis, is contaminated by the confusion with capital services and the capital stock. The purpose of this paper is to do a multi-sector version of the Hayashi-Prescott one-sector exercise. Besides government, there are two market sectors non-it and IT sectors and the household sector producing service flows from owner-occupied housing and consumer durables. By definition, the TFP growth for household and government is zero. In Section 2, we present our multi-sector accounting system matching this multi-sector model. Our data shows, not surprisingly, that the TFP growth in the IT sector is much higher than that in the non-it sector. We confirm in Section 3 that the TFP growth rate by the macro growth accounting is indeed higher than the genuine TFP growth, which a weighted average of the sectoral TFP growth rates. However, the two TFP growths show similar movements, with a marked decline in the 1990s. Section 4 presents the multi-sector growth model. It distinguishes between non-it capital and IT capital in the production function. In Section 5, following Hayashi and Prescott (2002), we calibrate the model to the Japanese economy in the period and report results from the simulation of the model from year 1990 on. The multi-sector model does well in accounting for the output slump of the 90s but less well than the one-sector model for the rise in the capital- 3

5 output ratio in the 1990s. The model s prediction about the long-run output growth depends on how much resources are directed to the IT sector. If the allocation of hours between non-it and IT sectors remains at the (97.2%, 2.8%) breakdown of year 2000, the long-run percapita GDP growth rate is 1.1%. To raise it to 2.0%, the private labor allocation must move in favor of IT, from (97.2%, 2.8%) to (90%, 10%). Section 6 is a brief conclusion. 2. The Multi-Sector Accounting Framework The theoretical model to be presented later in the paper is a multi-sector model with two market sectors (non-it and IT goods-producing sectors) and two non-market sectors (the household and government sectors). This section describes how the model s empirical counterpart, a multi-sector accounting system, is constructed. Its production account is derived from the 47- sector system developed in Jorgenson and Nomura (2005) (hereafter, JN). 2 Its final demand components, too, are from the KEO Database. Output and Value Added Table 1: Value Added at Factor Costs market production non-market production non-it sector (j = 1) IT sector (j = 2) household sector (j = H) gov t sector (j = G) value added P 1t Y 1t P 2t Y 2t P Ht Y Ht P Gt Y Gt non-it capital cost P 1t r 11t K 11t P 2t r 12t K 12t P Ht r 1dt K 1Ht P Gt r 1Gt K 1Gt IT capital cost P 1t r 21t K 21t P 2t r 22t K 22t P Ht r 2dt K 2Ht P Gt r 2Gt K 2Gt labor cost W 1t L 1t W 2t L 2t W Gt L Gt 2 It builds on the 43-sector KEO Database, which is a comprehensive productivity database for the Japanese economy maintained at the Keio Economic Observatory (KEO), Keio University, Japan. It consists of a time-series of inputoutput tables and detailed inputs of capital and labor. See Kuroda, Shimpo, Nomura, and Kobayashi (1997) for a detailed documentation. From the 43 industries in the KEO database, JN separates out three IT producing industries computers, communications equipment, and electronic components to form 47 industries. 4

6 By way of establishing the notation, Table 1 shows value added (also called net output in the productivity literature) at producer prices and their breakdown into factor costs for the four sectors. We define the IT sector (j=2) as consisting not only of the three IT industries in JN s 47 industries (which are computers and peripherals, communications equipment, and electronic components), but also of computer software. The software sector is defined narrowly as computer programming and other software services: custom software, pre-packaged software, own-account software, games, and other software, excluding data processing and other related information services. 3 Real value added of the IT sector, Y 2t, is the translog index of value added of these four industries. 4 Similarly, we define real value added of non-it sector Y 1t as the translog index of value added of all the industries except the IT sector, household, and government sectors. The price indexes of value added at producer prices, P jt (j = 1, 2), are derived by dividing the aggregated nominal value added by real value added in each sector. As in JN (Jorgenson and Nomura (2005)), the household sector produces rental service of owner-occupied housing and consumer durables with no labor input. Those rental services are consumed by the household sector itself, by definition. Nominal production, P Ht Y Ht, equals factor costs, which consist entirely of the user costs of owner-occupied housing and consumer durables. The government sector produces government service that is consumed by the government itself, by definition. The imputed nominal value of government services, P Gt Y Gt, is defined 3 We estimate the output and inputs of computer software sector as follows. In the Japanese 2000 benchmark inputoutput table produced by Statistics Bureau, Ministry of Internal Affairs and Communications, production activity of software sector is not divided from information services (851201), although the commodity of software ( ) is separated. We estimate production of the software sector using the activity in The own-account software is not included in the production of software ( ) of the Japanese 2000 benchmark IO table and still is not capitalized in the Japanese National Accounts by Economic and Social Research Institute (ESRI), Cabinet Office. For output and inputs of own-account software, we use the estimates in Nomura (2004b). 4 In general, let Y jt and P jt be real value added and the associated price index of sector j in period t. The translog quantity index Y t is defined as ln Y t = v jt ln Y jt, j where ln Y t ln Y t ln Y t 1 and v jt is the two-period average share of sector j s nominal value added in total nominal value added. 5

7 as capital costs and the value of labor input, where capital costs include not only consumption of public capital but also total capital service cost of publicly owned capital. Table 2 report sectoral shares of nominal value added. The imputed prices of value added P jt for those two nom-market sectors (j=h,g) are defined by dividing nominal value added by the quantities Y jt (j=h,g), which are defined in Equation (2.5) below. 5 Table 2: Value Added Shares private G Total sectoral breakdown of private non-it (v Y 1t ) IT (vy 2t ) H (vy Ht ) (0.8) (1.0) (1.6) (1.9) (1.8) (2.0) Note: Shares in percents. Values in parentheses represent shares of IT producing sector excluding software. Various capital assets can be divided into two groups, depending on their sectoral origin. The non-it capital or asset 1 is those assets produced in the non-it sector, while the IT capital or asset 2 are produced in the IT sector. 6 Since value added is at factor costs, it can be divided into payments to asset 1 (non-it capital), asset 2 (IT capital), and labor. We use v K for the factor share ijt 5 In JN, the government and household sectors have intermediate inputs to produce their non-market services. In this paper, to simplify the production functions, we treat these intermediate inputs as government consumption and household consumption of non-it good (i=1) at the final demand, respectively, so that there is no intermediate inputs for these sectors. 6 In JN, the most detailed asset classification has 102 assets. Of those 102 assets, to reflect the definition of the IT sector mentioned above, the IT capital in this paper is composed of electronic computer and peripherals, wired communication equipment, wireless communication equipment, other communication equipment, custom software, pre-packaged software, and own-account software. 6

8 of asset i (i = 1, 2) and v L jt for the labor share in sector j (j = 1, 2, H, G). So vk 1jt + vk 2jt + vl jt = 100%. Data on factor cost shares are in Table 3. Table 3: Factor Cost Shares non-it IT H G v K 11t v K 21t v L 1t v K 12t v K 22t v L 2t v K 1Ht v K 2Ht v K 1Gt v K 2Gt v L Gt Note: Shares in percents. v K ijt and vl are the cost shares of capital and labor, respectively. jt Capital and Labor Each sector utilizes many different capital assets. At the most detailed level of asset classification, the real capital stock and capital services are identical. However, when assets are aggregated into broader classes, the two are not the same, as first pointed out by Jorgenson and Griliches (1967). The real capital stock is the simple sum (valued at some base year prices) of those assets that belong to the broader asset class in question, while capital services aggregated over those assets is an index (e.g., the translog index) constructed from the user costs and the real capital stocks of those assets. The user costs of capital fully reflect the heterogeneity in productivity within the broader asset class. For the 47 sectors, JN calculated the real capital stock and the translog index of capital services for non-it and IT assets. 7 Those quantities are aggregated into our four broader sectors (j = 1, 2, H, G) to obtain the real capital stock and the translog capital services index for the 7 The user costs in JN incorporate the Japanese tax structure. The detailed formula is given in Nomura (2004a, ch.3), where he considers capital consumption allowance, income allowance and reserves, special depreciation, corporate income tax, business tax, property tax, acquisition taxes, debt/equity financing, and personal taxes on capital gain and dividend. Nomura (2004a, ch.3) measures effective tax rates and tax wedges, based on estimated before-tax and after-tax rates of return. This estimate gives the effective tax rate for capital income τ kt in our model. 7

9 non-it and IT assets. For asset i (i=1 for non-it or 2 for IT) in sector j in period t, we use K ijt for the real capital stock and K for the capital services index.8 The ijt ratio Q K ij K ijt /K ijt, which converts real capital stock into capital services, is called the capital quality in the productivity literature. It captures the heterogeneity within the asset class mentioned above. Going back to Table 1, the rental rate r ijt is defined to satisfy the relationship P jt r ijt K ijt = nominal value of capital services for asset i in sector j. (2.1) By definition, for any sector j (j = 1, 2, H, G), the IT capital stock K 2jt and IT capital services K 2jt do not include land. Land would be in non-it capital, but we do not include land in K 1jt and K ; see below for how we account for land as non-it capital. Despite the exclusion of land 1jt from K 1jt, the rental rate r 1jt of non-it capital includes land rent. The labor equivalent of real capital stock is total hours worked. Labor input differs from hours worked because it is an index of labor inputs of various kinds, distinguished by worker characteristics such as education. JN calculated total hours worked and the translog index of labor input for the 47 industries. 9 We can aggregate those quantities into our four broader sectors (j = 1, 2, H, G). We use L jt for total hours worked in sector j and L for the translog index of jt labor services. The labor quality Q L jt (j=1,2,g) is L jt /L jt. Nominal hourly wage rate in sector j, W jt, is defined to satisfy W jt L jt = nominal labor costs in sector j. (2.2) 8 JN use symbol K for capital services and Z for the real capital stock. 9 JN utilizes the KEO Database, which classifies workers by sex, age (eleven classes), educational attainment (four classes for males, three classes for females), employment class (three types: employees, self-employed, and unpaid family workers), and industry (forty-three). See Kuroda, Shimpo, Nomura, Kobayashi (1997, ch.4) for more detail. 8

10 Table 4: Allocation of Capital and Labor hours worked: L jt non-it capital stock: K 1jt IT capital stock: K 2jt private G private G private G breakdown breakdown breakdown non-it IT non-it IT H non-it IT H Note: Shares in percents. K 1jt excludes land. Allocation of the capital stock (excluding land) and total hours among sectors are reported in Table 4. Productivity To account for the role of land in the measurement of TFP (total factor productivity), we introduce a variable, ϕ jt, which converts the translog capital services index without land, K, into one 1jt with land for sector j. For each sector, we can define two measures of TFP growth, one (the pseudo TFP growth) that takes neither the capital and labor quality nor land into account (v T jt ) and the one (the genuine TFP growth) that does (v T ). They are defined as jt v T jt = ln Y jt v K ijt ln K ijt v L jt ln L jt, v T jt i=1,2 = ln Y jt i=1,2 (2.3a) v K ijt ln ( Q K ijt K ijt) v K 1jt ln ϕ jt v L jt ln ( Q L jt L jt), (2.3b) where v K ijt and vl are the two-period average cost shares of capital and labor in value added. So jt the genuine TFP growth v T jt is related to its pseudo cousin v T jt as v T jt = v T jt v K ijt ln QK ijt vl jt ln QL jt vk 1jt ln ϕ jt. (2.4) i=1,2 9

11 Both versions of TFP growth recognize the heterogeneity in productivity between the non-it and IT capital. The genuine TFP growth further takes into account the heterogeneity within each broad asset class. It also incorporates the labor heterogeneity and the contribution of land. Quantities of services produced by the household and government sectors are defined as the translog index of factor inputs in sector j (j=h, G): ln Y jt = i=1,2 where labor input in the household sector is zero: L Ht = 0. Table 5 reports two measures of TFP growth, v T jt v K ijt ln ( Q K ij K ijt) + v L jt ln ln ( Q L jt L ) jt + v K 1jt ϕ jt, (2.5) and vt, and aggregate TFP growth for the jt privete sector. By construction, the genuine TFP growths for the household and government sectors are zero. The pseudo TFP growths, v T (j=h,g) differ from 0, thanks to the exclusion of jt quality change and land. For the two market sectors (non-it and IT sectors), Figure 2 displays the genuine TFP growth v jt t (j = 1, 2). The rapid IT productivity growth is in stark contrast to the stagnation of the non-it sector. As noted in Jorgenson and Nomura (2005), the IT growth has accelerated in the second half of the 1990s. The dotted line in the Figure is the TFP growth of the IT sector when software is not included. The inclusion of software, while raising the value-added share of the IT sector in the 1980s and after as shown in Table 2, pulls down the TFP growth. 10

12 Table 5: TFP Growth: Two Measures pseudo TFP genuine TFP private G private by sector (v T jt ) by sector (vt jt ) private GDP non-it IT H non-it IT (16.3) (16.4) (16.2) (15.9) (12.1) (11.1) (11.2) (10.5) (4.3) (3.8) (18.1) (17.5) Note: Average annual growth rate in percents. Values in parentheses is TFP growth rates for IT producing sector excluding software. Land is excluded from non-it capital in the definition of v T jt and included in vt jt. Private TFP growth is the weighted average, over j = 1, 2, H, of the sectoral TFP growths, with the weights given by the value-added shares in Table 2. Private GDP growth is the weighted average, over j = 1, 2, H, of the sectoral real value-added growths, with the weights given by the value-added shares in Table 2. Growth rates in all calculations are log differences non-it producing sector IT producing sector IT producing sector excluding software Figure 2: TFP Growth Rate: Non-IT and IT Producing Sectors 11

13 Consumption and Investment The components of domestic final demand are in Table 6. Household consumes non-it goods (C 1Ht ) and own-produced rental services of consumer durables and owner-occupied housing (Y Ht ). C 1Ht includes intermediate inputs to household production Y Ht. Similarly, C 1Gt includes intermediate inputs to government production of services Y Gt. The output of the IT sector is not consumed. Table 6: Domestic Final Demand Consumption Investment H G non-it ind. IT ind. H G non-it goods P 1t C 1Ht P 1t C 1Gt P 1t I 11t P 1t I 12t P 1t I 1Ht P 1t I 1Gt IT goods P 2t I 21t P 2t I 22t P 2t I 2Ht P 2t I 2Gt Hou. service P Ht Y Ht Gov. service P Gt Y Gt Our estimate of consumption is based on the time-series of input-output tables in the KEO Database. To define the value at before-tax prices, we deducted net indirect tax from C 1Ht. The consumption tax rate τ ct is calculated as the ratio of the total value of net indirect tax to P 1t C 1Ht. Investments are calculated from the fixed capital formation matrix in Nomura (2004a). Physical depreciation rates δ ijt for asset i in sector j are aggregated from the depreciation rates for 95 produced assets and 16 consumer durables used in JN. For each asset i (i = 1, 2), the depreciation rate can differ across sectors and over time because the asset composition within each sector varies. Nevertheless, the aggregated depreciation rates reported in Table 7 are fairly uniform across sectors, except for the non-it asset of government, which includes infrastructure. 12

14 Table 7: Depreciation Rates non-it capital: δ 1jt IT capital: δ 2jt non-it IT H G non-it IT H G Note: in annual percentages. 3. Aggregation over Sectors and Assets We have calculated, in Table 5, the TFP growth for each sector and the private sector as a whole that takes into account the heterogeneity between the two assets (non-it and IT capital). In contrast, the growth accounting in the macro literature (see, e.g., Klenow and Rodriguez-Clare (2001) and Hayashi and Prescott (2002) (hereafter HP)) is so wedded to the one-sector paradigm that the aggregate capital input is defined simply as the value of nominal capital stock deflated by the output deflator. In this section, we apply this macro growth accounting to the multi-sector dataset described in the previous section. For real private GDP (real aggregate value added) at factor costs, we use the translog index. That is, let (P jt, Y jt ) be the value-added deflator and real value added in sector j (j = 1, 2, H) from our multi-sector dataset. We calculate aggregate real value added Y t as the translog index over industry value added. 10 The growth rate of private real GDP has already been reported in Table 5. The implicit GDP deflator is defined as the ratio of nominal aggregate value-added to Y t. The solid line in Figure 3 is real GDP per working-age population (the number of persons aged 20-69) thus calculated from our multi-sector system, detrended at 2% (which has been the 10 We also calculated the Fisher chain index and found it to be virtually identical to the translog index. 13

15 aggregated from the multi-sector accounting system SNA Figure 3: Real Private GDP Relative to 2% Trend long-run growth rate for the leader country (the U.S.) over the past century) and normalized to 100 for It shows that real private GDP Y t grew much faster than 2% until 1991 but slower than 2% thereafter. Just to show that the basic picture remains the same, the Figure also shows, in the dotted line, the similarly detrended official real GDP per worker from the Japanese national accounts (on the SNA93 basis). 11 There are a number of definitional differences between our GDP and the official Japanese SNA93 GDP: the official SNA93 GDP includes the government sector while our GDP doesn t; our GDP is a (translog) chain index while the official GDP is a fixed-weight Paasche index; 12 service flows from consumer durables are included in our GDP, and so forth. Compared to the official GDP, our measure shows slightly less growth in the 1980s and a slightly severer slump in the 1990s. Another macro variable of interest is the capital-output ratio. The macro growth accounting typically uses the GDP deflator to convert nominal into real. That is, the aggregate capital stock K t is defined as the ratio of the total nominal value of the capital stocks in sectors 1, 2, and H to the GDP deflator defined above. So the capital-output ratio K t /Y t equals the ratio of private 11 The SNA refers to the System of National Accounts. The SNA93 is a set of rules and standard for national accounting recommended by the United Nations in Currently, the official SNA93 chain index is available only since

16 3.4 aggregated from the multi-sector accounting system SNA Figure 4: Capital-Output Ratio nominal capital stock to nominal output: K t P 1t j=1,2,h K 1jt + P 2t j=1,2,h K 2jt, (3.1) Y t j=1,2,h P jt Y jt where K ijt is the capital stock of asset i in sector j and P it is the deflator for value added in sector i (see Table 1). Figure 4 shows, in the solid line, the capital-output ratio thus calculated from our multi-sector dataset. It shows a 25% rise from 2.62 in 1990 to 3.28 in The official capital-output ratio, with the capital stock as well as nominal GDP from the Japanese SNA93, is the dotted line. It is much lower and shows a much slower rise in the 1990s. The most important reason for the difference is that the depreciation rates in our multi-sector accounting system are lower. 13 The macro growth accounting as practiced in HP (Hayashi and Prescott (2002)) is based on the following identity: Y t = A 1/(1 θ) t N t ( Kt Y t ) θ/(1 θ) ( Lt N t ), (3.2) where N t is the working-age population (persons aged 20-69) and A t is the macro TFP defined 13 Nomura (2004a, Chapter 2) argues that the depreciation rates in the Japanese SNA are too high. Other reasons include the following. Our measure of nominal capital stock does not equal the nominal capital stock valued at investment goods prices. Own-account software and pre-packaged software are included in our measure of the capital stock. 15

17 to satisfy the aggregate Cobb-Douglas production function: Y t = A t K θ t L 1 θ t. (3.3) Our measure of aggregate labor L t is aggregate hours worked in the three private sectors. Table 8 shows the HP growth accounting applied to our multi-sector dataset with θ = (the capital share parameter used in HP). Despite the substantial differences in the definition, the overall picture is the same as in HP: both per capita output growth and the TFP growth slowed down to less than 1%, the capital-output ratio increased, and labor input declined in the 1990s. Comparing the macro TFP growth (the growth rate of A t ) in this table to the aggregate TFP growth reported in Table 5 (see the column labeled private for either the pseudo TFP or the genuine TFP), we see that the macro estimate is biased upward, although the movement over time is similar. period Table 8: Macro Growth Accounting Y t N t A t A 1/(1 θ) t K t Y t ( Kt Y t ) θ/(1 θ) [4.1] [0.7] [2.4] [0.4] [0.3] [0.4] L t N t Note: Growth rates in annual percentages. N t is the working-age population (number of persons aged 20-69). Y t is chained real private GDP, L t is total hours worked in the private sector, K t is real capital stock, and A t is macro TFP. See text for the precise definition of Y t, K t, L t, A t. θ = The numbers in brackets are the aggregate TFP growth rates in Hayashi and Prescott (2002), calculated from the worksheet downloadable from hayashi/hp/fed_data.xls. 16

18 4. The Two-Sector Growth Model with Consumer Durables This section presents our theoretical model to be confronted with the data from the multi-sector accounting system. Main features of the model are the following. Consistent with the multi-sector accounting system, there are two market-provided goods (non-it goods and IT goods) and two non-market goods (government services and household services). The IT goods are not consumed. Government services do not enter the household s utility function. The two market goods (non-it and IT goods) are internationally tradable. We assume that the country is small enough not to influence the relative price of IT goods in terms of non-it goods. This means that the relative price is exogenous to the model. Unlike in HP (Hayashi and Prescott (2002)), where the capital stock includes claims on the rest of the world, we separate external assets from domestic capital. The time path of external assets is treated as exogenous. This means that both net income from abroad and net exports are exogenous to the model. 14 Unlike in HP, labor supply is exogenous. The sectoral allocation of total labor, too, is exogenous. We are forced to make this assumption because the market equilibrium is one of near-complete specialization under the observed relative price if labor is allowed to move freely between sectors. We now turn to a more detailed description of the model. Households The stand-in household s utility function is β t N t u(c 1t, d t ), β (0, 1), (4.1) t=0 where N t is working-age population, c 1t is per-worker consumption of good 1 (the non-it good), and d t is the service flow (from owner-occupied housing and consumer durables) produced by 14 Recall from national income accounting that the net increase in external assets equals the current account and that the current account is the sum of next exports (the trade balance) and net income from abroad. 17

19 the household. If K it is the quantity of private capital stock i obtained from investment in good i (i = 1, 2), the household sets aside µ it N t d t of it as capital input for producing N t d t units of household services. These capital input requirement coefficients (µ 1t, µ 2t ) depend on the rental rates of capital (see below). We allow two distorting taxes, the consumption tax (the tax rate: τ ct ) and the tax on capital income (τ kt ). Labor supply is exogenous to the model, so the tax on labor is not distortionary and is lumped into the lump-sum tax τ ht. The household s budget constraint is (1 + τ ct )N t c 1t + [K 1,t+1 (1 δ 1 )K 1t ] + P t [K 2,t+1 (1 δ 2 )K 2t ] + (FA t+1 FA t ) w 1t L 1t + w 2t L 2t + w Gt L Gt + [r 1t (K 1t µ 1t N t d t ) + P t r 2t (K 2t µ 2t N t d t )] (4.2) τ kt [(r 1t δ 1 )(K 1t µ 1t N t d t ) + P t (r 2t δ 2 )(K 2t µ 2t N t d t )] τ ht + NI t, where P t is the relative price (the price of good 2 in terms of good 1), δ i is the depreciation rate of asset i, FA t is foreign assets in terms of non-it goods (good 1), w jt is the wage rate measured in good 1 paid by sector j, L jt is labor supply to sector j, r it is the rental rate for asset i (so the rental price in terms of good 1 of asset 2 equals P t r 2t ), and NI t is net income from abroad measured in good In terms of the notation of Tables 1 and 6, P t = P 2t /P 1t, c 1t = C 1Ht /N t, d t = Y Ht /N t, w jt = W jt /P 1t, K i,t+1 (1 δ i )K it = I i1t + I i2t + I iht (i = 1, 2), and µ it N t d t = K iht (i = 1, 2). In the data, the rental rate r ijt depends on sector j because of the asset heterogeneity within the broader asset classes of non-it and IT capital, but in the model, which does not recognize this heterogeneity, the rental rate does not depend on j. Let β t Λ 1 t be the Lagrange multiplier. Being the reciprocal of the shadow price of the budget constraint, Λ t measures the wealth of the household. The FOCs (first-order conditions) for 15 This budget constraint implies that the household does not pay the consumption tax on purchases of non-it and IT goods to be used for household production. If this counterfactual assumption is to be avoided, we would have to treat those flow of durables purchases separately from investments in the capital stock to be rented out to firms, but that adds another co-state variable to the system. Our numerical procedure for computing the perfect foresight equilibrium path cannot handle more than one co-state variable. 18

20 optimality are: u c1 (c 1t, d t ) = (1 + τ ct )Λ 1 t, (4.3) u d (c 1t, d t ) = P dt Λ 1 t, (4.4) β[1 + (1 τ k,t+1 )(r 1,t+1 δ 1 )] = Λ t+1 Λ t, (4.5) β[1 + (1 τ k,t+1 )(r 2,t+1 δ 2 )] = P t P t+1 Λ t+1 Λ t, (4.6) where P dt [(1 τ kt )r 1t + τ kt δ 1 ]µ 1t + P t [(1 τ kt )r 2t + τ kt δ 2 ]µ 2t (4.7) is the imputed price of household services in terms of sector 1 output. 16 The user costs are (1 τ kt )r it + τ kt δ i rather than r it, because household production is not taxed. (4.5) and (4.6) yield (Euler equation) (arbitrage) Λ t+1 = β[1 + (1 τ k,t+1 )(r 1,t+1 δ 1 )], Λ t (4.8) P t+1 = 1 + (1 τ k,t+1)(r 1,t+1 δ 1 ) P t 1 + (1 τ k,t+1 )(r 2,t+1 δ 2 ). (4.9) The first of these two equations will be referred to as the Euler equation because it describes how the household wealth Λ t evolves over time. The second equation will be called the arbitrage condition governing the portfolio of the two assets K 1t and K 2t. We can solve (4.3) and (4.4) for (c 1t, d t ) as a function of the consumption tax rate τ ct, the imputed price (P dt ), and the shadow price Λ t : (Frisch demands) c 1t = c 1 (τ ct, P dt, Λ t ), d t = d(τ ct, P dt, Λ t ). (4.10) This demand system, with the shadow price Λ replacing income, is called Frisch demands. Use of Frisch demands enforces the household FOCs (4.3) and (4.4) in the equilibrium conditions. The input requirement coefficients (µ 1t, µ 2t ) for producing household services can be made endogenous, as the solution to the cost minimizing problem: 17 min P dt s.t. F d (ϕ Ht Q K µ 1t,µ 1Ht µ 1t, Q K 2Ht µ 2t) = 1, (4.11) 2t 16 It equals P Ht /P 1t in the notation of Section The constraint in the problem can also be written as F d (ϕ Ht Q K 1Ht µ 1tN t d t, Q K 2Ht µ 2tN t d t ) = N t d t due to constant returns to scale. 19

21 where F d is a linear homogeneous production function for household services, Q K (i = 1, 2) is iht the capital quality defined in Section 2 to capture the asset heterogeneity, and ϕ Ht is the factor accounting for land in asset 1. By definition, there is no TFP growth in household production, so the production function is stationary. We write the solution as ( ) ( ) (1 τ kt )r 1t + τ kt δ 1 µ 1t = µ 1 /(ϕ Ht Q K1Ht P t [(1 τ kt )r 2t + τ kt δ 2 ] ), µ (1 τ kt )r 1t + τ kt δ 1 2t = µ 2 /Q K 2Ht P t [(1 τ kt )r 2t + τ kt δ 2 ]. (4.12) Firms The technology of the two market sectors is described by the constant-returns-to-scale production functions Y 1t = Y 1 (ϕ 1t Q K 11t K 11t, Q K 21t K 21t, Q L 1t L 1t; A 1t ), (4.13) Y 2t = Y 2 (ϕ 2t Q K 12t K 12t, Q K 22t K 22t, Q L 2t L 2t; A 2t ), (4.14) where K ijt is the amount of privately-held asset i rented by sector j in date t and A is the level of jt technology. The capital quality Q K ijt converts the capital stock into capital services, QL jt measures the quality of labor in sector j, and the factor ϕ jt accounts for land. The FOCs are r 1t = Y 1t K 11t, P t r 2t = Y 1t K 21t, r 1t = P t Y 2t K 12t, r 2t = Y 2t K 22t, w 1t = Y 1t L 1t, w 2t = P t Y 2t L 2t. (4.15) The Government The government collects taxes to finance government expenditure in goods and services (the sum of government consumption and investment) in two goods, G 1t and G 2t, and payments to hire labor L Gt used for producing government services. The lump-sum tax τ ht is adjusted to meet the government budget constraint. In terms of the notation of Table 6, G 1t = C 1Gt + I 1Gt and G 2t = I 2Gt. Neither the government production function nor breakdown of government expenditure between consumption and investment needs to be specified, because it does not affect the equilibrium of the private sector. 20

22 Market Equilibrium The market equilibrium conditions are: (asset 1) K 11t + K 12t + µ 1t N t d t = K 1t, (4.16) (asset 2) K 21t + K 22t + µ 2t N t d t = K 2t, (4.17) (RC) N t c 1t + [K 1,t+1 (1 δ 1 )K 1t ] + P t [K 2,t+1 (1 δ 2 )K 2t ] = (Y 1t G 1t ) + P t (Y 2t G 2t ) NX t. (4.18) Here, the label RC stands for resource constraint and NX t is net exports in terms of good 1. Equilibrium We can now define competitive equilibrium. Take as given: a government policy {K 1Gt, K 2Gt, L Gt, G 1t, G 2t, τ ct, τ kt } t=0, labor supply to each sector {L 1t, L 2t } t=0, external assets {FA t } t=0 and net exports {NX t} t=0, the relative price {P t } t=0, the capital and labor quality and the land conversion factor {Q K ijt, QL jt, ϕ jt} t=0 (i = 1, 2; j = 1, 2, H, G), the technology level {A 1t, A 2t } t=0. A competitive equilibrium given an initial condition (K 10, K 20 ) is a sequence of factor prices, {r 1t, r 2t, w 1t, w 2t } t=0, the household wealth {Λ t} t=0, and associated quantities, {K 1,t+1, K 2,t+1, K 11t, K 12t, K 21t, K 22t }, such that the Euler equation (4.8), the arbitrage condition (4.9), the firm s FOCs t=0 (4.15), the market equilibrium conditions ((4.16)-(4.18)) are satisfied, where (Y 1t, Y 2t ) in those conditions are given by (4.13) and (4.14), c 1t in the RC (resource constraint) and d t in the asset market equilibrium condition are given by (4.10), and (µ 1t, µ 2t ) in the asset market equilibrium condition are given by (4.12). In this definition, the government budget constraint is not an equilibrium condition, because the lump-sum tax τ ht is assumed to meet the constraint. The household budget constraint is 21

23 not included, because it is implied by the government budget constraint, the factor exhaustion condition that value added equals factor payments (an implication of the linear homogeneity of the production function and the marginal productivity conditions), the market equilibrium conditions, and the identity that the increase in external assets, FA t+1 FA t, equals net exports NX t plus net income from abroad NI t. Implications of The Cobb-Douglas Technology In what follows, we assume the Cobb-Douglas form for the production functions. So (4.13) and (4.14) can be written as Y 1t = A 1t Y 2t = A 2t ( ϕ1t Q K 11t K ) θ11 ( 11t Q K 21t K ) θ21 ( 21t Q L 1t L ) 1 θ11 θ 21 1t = A 1t K θ 11 11t Kθ 21 21t L1 θ 11 θ 21, 1t (4.19) ( ϕ2t Q K 12t K ) θ12 ( 12t Q K 22t K ) θ22 ( 22t Q L 2t L ) 1 θ12 θ 22 2t = A 2t K θ 12 12t Kθ 22 22t L1 θ 12 θ 22, 2t (4.20) where A jt A jt ( ϕjt Q K 1jt ) θ1j ( ) θ2j ( 1 θ1j θ Q K 2jt Q L 2j jt), j = 1, 2. (4.21) This shows that, for the Cobb-Douglas technology, the production function can be defined for the capital stocks, with an appropriate re-definition of the TFP term. The growth rate of A is the jt genuine TFP growth rate, v T, of Section 2. The above expression of the production function jt makes it clear that what matters for equilibrium is the pseudo TFP growth rate, v T jt, which equals the growth rate of A jt defined here. (4.11): We will also assume the Cobb-Douglas form for the household production function F d in F d (ϕ Ht Q K 1Ht K 1Ht, Q K 2Ht K 2Ht) = A H ( ϕht Q K 1Ht, K 1Ht ) γ ( Q K 2Ht K ) 1 γ 2Ht = AHt K γ 1Ht K1 γ 2Ht, (4.22) where A Ht = A H ( ϕht Q K 1Ht The utility function u(c 1t, d t ) is linear logarithmic: ) γ ( Q K 2Ht) 1 γ. (4.23) u(c 1t, d t ) = µ log(c 1t ) + (1 µ) log(d t ). (4.24) With these functional-form assumptions for household, the Frisch demands (4.10) and the de- 22

24 mand for capital inputs for household production (4.12) become (Frisch demands) c 1t = µλ t 1 + τ ct, d t = (1 µ)λ t P dt, (4.25) P dt = γ γ (1 γ) (1 γ) ((1 τ kt )r 1t + τ kt δ 1 ) γ (P t [(1 τ kt )r 2t + τ kt δ 2 ]) 1 γ /A Ht, (4.26) γ K 1Ht µ 1t N t d t = (1 µ)n t Λ t, (1 τ kt )r 1t + τ kt δ 1 1 γ K 2Ht µ 2t N t d t = P t [(1 τ kt )r 2t + τ kt δ 2 ] (1 µ)n tλ t. (4.27) Thanks to the unit elasticity in both the demand for and the supply of household services, a change in A Ht does not affect the demand for assets µ it N t d t in household production. Combining (4.22), (4.26), and (4.27), we obtain P dt Y Ht = P dt A Ht K γ 1Ht K1 γ 2Ht = (1 µ)n tλ t, (4.28) which states that the value (in terms of good 1) of household production does not depend on the pseudo TFP A Ht. Detrending With the Cobb-Douglas technology and the linear-logarithmic preferences, it is possible to transform the system so that it involves only the growth rates, but not the levels, of the TFPs. To this end, define two trends: 1 θ 22 v X 1t A1t θ 21 v A2t θ 12 v N t, X 2t A1t 1 θ 11 v A2t N t with v 1 θ 11 θ 22 + θ 11 θ 22 θ 12 θ 21. (4.29) Define also lower-case letters as ratios to these trends: k it K it X it (i = 1, 2), k ijt K ijt X it (i, j = 1, 2), l jt L jt N t (j = 1, 2), y jt Y jt X jt (j = 1, 2), p t P t ( ), λ t Λ t ( ). X1t X1t X 2t N t (4.30) 23

25 A very tedious algebra shows that the equilibrium conditions in terms of these detrended variables can be reduced to the following set of equations. ( X1,t+1 ) λ t+1 X 1t [ (Euler) ( ) = β 1 + (1 τk,t+1 )(r 1,t+1 δ 1 ) ], (4.31) λ Nt+1 t N t ( X1,t+1 ) p t+1 X 1t (arbitrage) p t ( X2,t+1 X 2t ) = (production FOCs) r 1t = θ 11 y 1t k 11t, p t r 2t = θ 21 (RC) 1 + (1 τ k,t+1 )(r 1,t+1 δ 1 ) 1 + (1 τ k,t+1 )(r 2,t+1 δ 2 ), (4.32) y 1t k 21t, r 1t = θ 12 p t y 2t k 12t, r 2t = θ 22 y 2t k 22t, y 1t = k θ 11 11t kθ 21 21t l1 θ 11 θ 21 1t, y 2t = k θ 12 12t kθ 22 22t l1 θ 12 θ 22 2t, (4.33) γ (asset 1) k 11t + k 12t + (1 µ)λ t = k 1t, (4.34) (1 τ kt )r 1t + τ kt δ 1 1 γ (asset 2) k 21t + k 22t + (1 µ)λ t = k 2t, (4.35) p t [(1 τ kt )r 2t + τ kt δ 2 ] ( ) X1t X 1,t+1 (z defined) z t+1 k 1,t+1 + p t ( ) k 2,t+1, (4.36) X2t X 2,t+1 X 1,t+1 X 1t z t+1 = (1 ψ 1t )y 1t + p t (1 ψ 2t )y 2t µ 1 + τ ct λ t + (1 δ 1 )k 1t + (1 δ 2 )p t k 2t ν t (y 1t + p t y 2t ). (4.37) Here, (ψ 1t, ψ 2t ) is the government share of output for each good and ν t is net exports-to-gdp (excluding household production) ratio: ψ it G it Y it (i = 1, 2) and ν t NX t /(Y 1t + P t Y 2t ). (4.38) The Transition Path and the Steady State From this set of equations, we can define a first-order dynamical system (k 1, k 2, λ), namely a mapping from (k 1t, k 2t, λ t ) to (k 1,t+1, k 2,t+1, λ t+1 ), as follows. i) Given (k 1t, k 2t, λ t ), use eight equations, (4.33)-(4.35) to solve for eight unknowns (r 1t, r 2t, k 11t, k 21t, k 12t, k 22t, y 1t, y 2t ). Intuitively, this enforces that the country s marginal rate of transformation between the two goods be equated to the world relative price. This step also gives (r 1t, r 2t ) as functions of (k 1t, k 2t, λ t ). Write them as: r it = r i (k 1t, k 2t, λ t ) (i = 1, 2). ii) Given (k 1t, k 2t, λ t, y 1t, y 2t ), use (4.37) to calculate z t+1. iii) Substitute r i,t+1 = r i (k 1,t+1, k 2,t+1, λ t+1 ) into (4.31) and (4.32). Given the value of z t+1 ob- 24

26 tained in the previous step, three equations (4.31), (4.32), and (4.36) can be solved for (k 1,t+1, k 2,t+1, λ t+1 ). The initial condition for the system is that the initial values for the two state variables (k 1t, k 2t ) are given. Given an appropriate initial value for the co-state variable λ t, we can generate the solution path using this mapping from (k 1t, k 2t, λ t ) to (k 1,t+1, k 2,t+1, λ t+1 ). This dynamical system is autonomous (i.e., the mapping from (k 1t, k 2t, λ t ) to (k 1,t+1, k 2,t+1, λ t+1 ) is stationary or time-invariant) if the forcing or exogenous variables l 1t, l 2t, X 1,t+1 /X 1t, X 2,t+1 /X 2t, N t+1 /N t, τ kt, τ ct, ψ 1t, ψ 2t, p t, and ν t are constant over time. The steady state or the equilibrium of this autonomous system can be determined uniquely from the above equations by dropping the time subscript, as follows. Dropping the time subscript in (4.31) pins down r 1, the steady-state value of r 1t. Given r 1, use the steady-state version of (4.32) to obtain r 2. Given r 1 and r 2, use (4.33) to calculate (k 11, k 21, k 12, k 22, y 1, y 2 ). Use the rest of the equations, (4.34)-(4.37) to pin down (k 1, k 2, z, λ). When we simulate the model in the next section, we will assume that the exogenous variables become constant either asymptotically (for l 1t, l 2t, ν t ) or after some fixed date (which is year 2000 in the simulation below), so the detrended system is asymptotically autonomous. The appropriate initial value of the co-state λ t is the one that guides the detrended system to approach the steady state in the long run (as time goes to infinity). This particular solution path satisfies relevant transversality conditions. Given the solution path for the detrended system, we can back out the equilibrium of the model before detrending using (4.30). This determines: Y 1t, Y 2t, K 1t, K 2t, K 11t, K 12t, K 21t, K 22t, r 1t, r 2t, Λ t. The value of household production, P dt Y Ht, can be determined from this by (4.28). All these endogenous variables are determined independent of the psudo TFP for the household sector, A Ht. The breakdown of P dt Y Ht between price and quantity does depend on A Ht and can be obtained from (4.26). The steady-state or the balanced growth path associated with the steady-state or the equilibrium of the detrended system has the following features. 25

27 The sectoral allocation of capital for each asset i (i = 1, 2) is constant, because the trend in K ijt does not depend on j. Obviously, Y 1t X 1t, Y 2t X 2t, P t ( P 2t P 1t ) X 1t X 2t, N t Λ t X 1t. (4.39) Thus if, as is the case in the calibrated model below, X 2t grows much faster than X 1t thanks to the rapid IT productivity growth, the relative price of IT goods declines rapidly and the growth of relative output level Y 2t /Y 1t is as rapid as the decline in the relative price. The sectoral nominal value-added shares in private GDP v Y jt = P jt Y jt P 1t Y 1t + P 2t Y 2t + P Ht Y Ht = Y 1t Y 1t +P t Y 2t +P dt Y Ht (for j = 1), P t Y 2t Y 1t +P t Y 2t +P dt Y Ht (for j = 2), (4.40) P dt Y Ht Y 1t +P t Y 2t +P dt Y Ht (for j = H), are constant. This is because, in addition to (4.39), we have P dt Y Ht X 1t by (4.28). It then follows from (4.26) that Y Ht X γ 1t X1 γ A 2t Ht, P dt ( P ( ) 1 γ Ht X1t ) A 1 Ht P 1t X. (4.41) 2t Let Y t be the translog index of real private GDP constructed from (P jt, Y jt ) (j = 1, 2, H) with P 2t = P 1t P t and P Ht = P dt P t for some arbitrary path of P 1t (which doesn t affect the index), and let v Y j be the steady-state value of the value-added shares discussed above. Then the steady-state growth rate of real private GDP is given by v Y 1 g X1 + v Y 2 g X2 + (1 v Y 1 vy 2 ) (γg X1 + (1 γ)g X2 + g H ), (4.42) where g Xj is the long-run growth rate of X jt and g H is the long-run growth rate of A Ht. From the firm s marginal productivity conditions, the wage ratio w 1t /w 2t under the Cobb- Douglas technology equals 1 θ 11 θ 21 Y 1t /L 1t 1 θ 21 θ 22 P t Y 2t /L 2t for all t. In the steady state, it equals w 1t w 2t in the steady state = 1 θ 11 θ 21 1 θ 21 θ 22 y 1 /l 1 py 2 /l 2, (4.43) where (p, l 1, l 2 ) are the constant long-run values of (p t, l 1t, l 2t ). It can be easily seen from tracing out the procedure for calculating the steady state explained above that y 1 /l 1 and y 2 /l 2 26

28 do not depend on (l 1, l 2 ) (an implication of the constant-returns-to-scale technology and the small country assumption). Thus the steady-state wage ratio between non-it and IT sectors does not depend on the allocation of labor between the two sectors. We close this section by commenting on our assumption that the relative price (P t P 2t /P 1t ), which is internationally given, is projected to grow at precisely the rate given by X 1t /X 2t, which is determined by the domestic TFP growth rates in non-it and IT sectors (see (4.29)). This is not as far-fetched as it might seem. As Jorgenson (2003) finds, the rapid decline in IT prices is common to G7 countries. If, as Jorgenson and Nomura (2005) argue, this relentless decline is rooted in developments in technology (primarily in semiconductors) that are widely understood by technologists and economists, it would not be too unrealistic to assume that the TFP growth rate in the IT sector in Japan is as high as it is elsewhere in the world. 5. Calibration and Results The detrended dynamical system described in the previous section has a set of parameters and exogenous variables. In this section, we calibrate the detrended system and specify the paths of exogenous variables. We will then discuss the steady-state properties of the system. The equilibrium of the model in levels can be obtained by solving the detrended system and then multiplying the solution by appropriate time trends. The last part of this section examines the transition dynamics thus calculated. As in Hayashi and Prescott (2002), the calibration is based on data for in principle, and the transition dynamics is set off from The presumption of this simulation exercise is that the representative agent learns, all of a sudden in 1990, about the paths of the exogenous variable from 1990 on. The prior expectations about the exogenous variable from 1990 on are whatever is consistent with the values of the endogenous variables of the model. 27